I. Q can be a perfect square.
II. Q is positive for all real values of x.
Select the answer using the code given below :
To solve the given problem, we need to analyze the equation \( Q = x^2 + bx + c \) under the condition that the sum of the roots is equal to the product of the roots.
We know that for a quadratic equation \( ax^2 + bx + c = 0 \) (in this case, \( a = 1 \) for \( x^2 + bx + c = 0 \)), the sum of the roots \( \alpha + \beta \) is given by \( -\frac{b}{a} \), and the product of the roots \( \alpha \beta \) is \(\frac{c}{a}\).
Given:
The sum of the roots = Product of the roots.
\(-\frac{b}{1} = \frac{c}{1}\)
This simplifies to:
\(b = -c\)
Now, let's analyze each statement:
- Statement I: Q can be a perfect square.
- Statement II: Q is positive for all real values of x.
Therefore, the correct choice is I only.
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